What are the flaws in the following reasoning?
By Gödel's theorem, for any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or statements $G_i(F)$) that cannot be derived from this formal system.
If we extend it to math in general (math is a formal system, that certainly includes ordinary arithmetic; we'll mark the whole math with $M$ here): in math ($M$) there exist true statements $G_i(M)$ that cannot be derived by any formal rules that currently exist in math. In other words, math in general cannot contain within itself an algorithmic procedure that will discover all possible true statements in math ($G_i(M)$).
Which basically means that there are true statements in math that need an external explorer to discover and prove them. We, people, are actually such explorers and we have the capability of finding and proving some of $G_i(M)$. (Well, it can be shown by using Gödel's theorem itself, but I'll omit this proof for brievity).
And finally, the above means that the world we live in cannot be described by math in all it's entirety. Because, if it could, our human mind (that is certainly a part of this world) would be also fully described by math, which would include an algorithmic description of our capability to find and prove $G_i(M)$, which is a contradiction.
Second take on the above reasoning (trying to be a bit more formal):
By Gödel's theorem, in any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or statements $G_i(F)$) that can be formulated in terms of $F$ but cannot be proven with $F$.
All current math (marked with $M$) is a formal system, to which Gödel's theorem is applicable. Hence there exist $G(M)$ (or multiple $G_i(M)$) that can be formulated with $M$, but cannot be proven by $M$.
Also, $M$ does not contain within itself an algorithmic procedure to even suggest $G(M)$, because the existence of this algorithmic procedure within $M$ would actually be a proof of $G(M)$. (More rigorous proof is required here, I guess.)
Humans (mathematicians) can certainly suggest some $G_i(F)$ and prove them afterwards using rules/approaches additional to $F$, thus forming a new formal system $F'$, where those $G_i(F)$ are proven to be true. (I need to provide some good examples here). Same holds for $M$.
Finally, currently known math $M$ cannot describe the world in all it's entirety, because it at least cannot describe our human capability to suggest $G_i(M)$. After we prove $G_i(M)$, we come up with new math $M'$. But $M'$ now contains $G_i(M')$ by Gödel's theorem, and we come back to step 3. This can go on infinitely many times, and neither $M$, nor $M'$, nor $M''$ etc. would be able to fully describe the world and, in particular, our mathematical thinking and the way we do math.
So, what are the flaws/mistakes in the above reasoning? We already have some good answers here, but maybe some additions/corrections?